A014950 -id:A014950 - cp's OEIS frontend

A232769 Numbers n not divisible by 9 such that n divides 10^n - 1 (A014950).

1, 3, 111, 4107, 151959, 5622483, 22494039, 208031871, 225121209, 832279443, 7697179227, 8329484733, 27486820443, 30794339391, 92366302683, 123199851603, 230915528769, 284795631399, 308190935121, 1017012356391

Offset: 1

Hans Havermann and Robert G. Wilson v, Nov 29 2013

nonn

The above terms reduced modulo 9 yield: 1, 3, 3, 3, 3, 3, 6, 3, 6, 6, 3, 6, 3, 6, 3, 3, 3, 3, 6, 3, 6, …, .

The only primes less than a billion which can divide members of this sequence are 3, 37, 5477, 607837, 1519591, 2028119, 15195911, 18235093, 44988079, 74202397, 247629013, 337349203, 395397319, 462411133, and 674699071. - Charles R Greathouse IV, Dec 03 2013

Ray Chandler, Table of n, a(n) for n = 1..55

Cf. A014950.

k = 3; lst = {1}; While[k < 10^10 + 1, If[ PowerMod[10, k, k] == 1, AppendTo[ lst, k]; Print@ k]; k += 3; If[ PowerMod[ 10, k, k] == 1, AppendTo[ lst, k]; Print@ k]; k += 6]; lst

is(n)=n%9 && Mod(10,n)^n==1 \\ Charles R Greathouse IV, Dec 03 2013

forstep(n=1,1e8,[2, 4, 4, 2, 4, 2, 2, 2, 6, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 6, 2, 2, 2, 4, 2, 4, 4, 2, 2], if(Mod(10,n)^n==1,print1(n", "))) \\ Charles R Greathouse IV, Dec 03 2013

a(22)-a(26) from Ray Chandler, Dec 11 2013

B-file extended through a(55) by Ray Chandler, Dec 24 2013

A127100 Numbers k such that k^2 divides 10^k-1.

Original entry on oeis.org

1, 3, 9, 111, 333, 3003003, 111111111, 225121209, 675363627, 27486820443, 32119664517, 82460461329, 24048075051027, 90180273183093, 225346555330209, 889778776887999, 3336670107774441, 10717272100393839, 19885751580714849, 27514334750263443

Offset: 1

Alexander Adamchuk, Jan 05 2007, Jan 07 2006

nonn

Subsequence of A014950.
First 7 terms are palindromes. a(n) is divisible by 3 for 1Alexander Adamchuk, Jan 25 2007
Except for 3, also numbers n such that the decimal expansion of 1/n^2 has period n. - Arkadiusz Wesolowski, Mar 13 2012

Cf. A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092, A014950.

Select[Range[20000], IntegerQ[(PowerMod[10, #, #^2 ]-1)/#^2 ]&]

More terms from Ryan Propper, Jan 06 2007

Further terms and edited by Max Alekseyev, May 09 2010

A177243 Numbers k such that k^3 divides 10^(k^2) - 1.

Original entry on oeis.org

1, 3, 9, 111, 333, 1467, 6813, 54279, 84573, 252081, 607947, 665667, 1001001, 1110519, 1823841, 3003003, 3129201, 12914001, 13785399, 22161609, 22957083, 37037037, 41089203, 49235049, 64021761, 92840751, 108503721, 111111111

Offset: 1

Alexander Adamchuk, May 06 2010

nonn

Cf. A127100 (k^2 divides 10^k-1), A014950 (k divides 10^k-1).

Join[{1},Select[Range[1112*10^5],PowerMod[10,#^2,#^3]==1&]] (* Harvey P. Dale, Sep 10 2019 *)

A014960 Integers n such that n divides 24^n - 1.

Original entry on oeis.org

1, 23, 529, 1081, 12167, 24863, 50807, 279841, 571849, 1168561, 2387929, 2870377, 6436343, 7009273, 13152527, 15954479, 26876903, 54922367, 66018671, 112232663, 134907719, 148035889, 161213279, 302508121, 329435831

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*24^(k-1) (cf. A014942).
All n > 1 in the sequence are multiple of 23. - Conjectured by Thomas Baruchel, Oct 10 2003; proved by Max Alekseyev, Nov 16 2019
If n is a term and prime p|(24^n - 1), then n*p is a term. In particular, if n is a term and prime p|n, then n*p is a term. The smallest term with 3 distinct prime factors is a(16) = 15954479 = 23 * 47 * 14759. - Max Alekseyev, Nov 16 2019

Max Alekseyev, Table of n, a(n) for n = 1..146

Prime factors are listed in A087807.
Cf. A014942.
Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014959 (b=22).

s = 1; Do[ If[ Mod[ s, n ] == 0, Print[n]]; s = s + (n + 1)*24^n, {n, 1, 100000}]
Join[{1},Select[Range[330*10^6],PowerMod[24,#,#]==1&]] (* Harvey P. Dale, Jan 19 2023 *)

More terms from Robert G. Wilson v, Sep 13 2000
a(9)-a(12) from Thomas Baruchel, Oct 10 2003
Edited and terms a(13) onward added by Max Alekseyev, Nov 16 2019

A014956 Positive integers k such that k divides 14^k - 1.

Original entry on oeis.org

1, 13, 169, 2041, 2197, 26533, 28561, 114413, 320437, 344929, 371293, 1487369, 4165681, 4484077, 4826809, 17962841, 19335797, 24355253, 50308609, 54153853, 58293001, 62748517, 77457601, 233516933, 249302027, 251365361, 316618289

Offset: 1

Olivier Gérard

nonn

Also, positive integers k such that k divides A014929(k).
13 divides a(n) for n > 1. All powers of 13 are terms. All a(n) that are not powers of 13 are divisible either by 157 or 677 or both. - Alexander Adamchuk, May 14 2010
Prime divisors of a(n) in order of appearance: {13, 157, 677, 11933, 122147, 52807, ...}. - Alexander Adamchuk, May 16 2010

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A128360.

Cf. A177914, A128394.

Join[{1}, Select[Range[2000000], PowerMod[14, #, #] == 1 &]] (* Robert Price, Mar 31 2020 *)

2 more terms from R. J. Mathar, Mar 05 2008
a(8)-a(23) from Alexander Adamchuk, May 14 2010
a(24)-a(44) from Alexander Adamchuk, May 16 2010
Edited by Max Alekseyev, Sep 10 2011

A066364 Prime divisors of solutions to 10^n == 1 (mod n).

Original entry on oeis.org

3, 37, 163, 757, 1999, 5477, 8803, 9397, 13627, 15649, 36187, 40879, 62597, 106277, 147853, 161839, 215893, 231643, 281683, 295759, 313471, 333667, 338293, 478243, 490573, 607837, 647357, 743933, 988643, 1014877, 1056241, 1168711, 1353173, 1390757, 1487867, 1519591, 1627523, 1835083, 1912969, 2028119, 2029759, 2064529

Offset: 1

Vladeta Jovovic, Dec 21 2001

nonn

			10^27-1 = 3^5*37*757*333667*440334654777631 is a solution to the congruence.

Max Alekseyev and Hans Havermann (Max Alekseyev to 501), Table of n, a(n) for n = 1..2060
Rüdiger Jehn and Kester Habermann, Properties of terms of OEIS A342810, arXiv:2106.05866 [math.GM], 2021.
Makoto Kamada, Factorizations of 11...11 (Repunit).

Cf. A014950, A001270, A027889, A007138, A114207.

fQ[p_] := Block[{fi = First@# & /@ FactorInteger[ MultiplicativeOrder[ 10, p]]}, Union[ MemberQ[ lst, #] & /@ fi] == {True}]; k = 4; lst = {3}; While[k < 180000, If[ p = Prime@ k; fQ@ p, AppendTo[ lst, p]; Print@ p]; k++]; lst (* Robert G. Wilson v, Nov 30 2013 *)

S=Set([3]); forprime(p=7,10^6, v=factorint(znorder(Mod(10,p)))[,1]; if(length(setintersect(S,Set(v)))==length(v), S=setunion(S,[p])) ); print(vecsort(eval(S))) \\ Max Alekseyev, Nov 16 2005

A prime p is a term iff all prime divisors of ord_p(10) are terms, where ord_p(10) is the order of 10 modulo p. - Max Alekseyev, Nov 16 2005

Edited by Max Alekseyev, Nov 16 2005

Edited by Hans Havermann, Jul 11 2014

A068382 Numbers k such that k divides 9^k - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 136, 160, 164, 200, 220, 250, 256, 272, 320, 328, 400, 440, 500, 512, 544, 550, 610, 640, 656, 680, 800, 820, 880, 1000, 1024, 1088, 1100, 1210, 1220, 1250, 1280, 1312, 1360, 1544, 1600, 1640, 1760

Offset: 1

Benoit Cloitre, Mar 05 2002

For all m the sequence includes 2^m, 10^m, 2*10^m, 10*2^m.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A014945, A014946, A014949, A014950.

Join[{1}, Select[Range[10000], PowerMod[9, #, #] == 1 &]] (* Robert Price, Apr 04 2020 *)

isok(n) = Mod(9, n)^n == Mod(1, n); \\ Michel Marcus, May 06 2016

A014957 Positive integers k that divide 16^k - 1.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 25, 27, 39, 45, 55, 63, 75, 81, 105, 117, 125, 135, 147, 155, 165, 171, 189, 195, 205, 225, 243, 273, 275, 315, 333, 351, 375, 405, 441, 465, 495, 507, 513, 525, 567, 585, 605, 609, 615, 625, 657, 675, 729, 735, 775, 819, 825, 855, 903

Offset: 1

Olivier Gérard

nonn

Also, positive integers k that divide A014931(k).

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A014956, A177805, A177807, A128358, A128360

Cf. A128396, A177916

Join[{1},Select[Range[1000],PowerMod[16,#,#]==1&]] (* Harvey P. Dale, Jun 12 2024 *)

A014957_list = [n for n in range(1,10**6) if n == 1 or pow(16,n,n) == 1] # Chai Wah Wu, Mar 25 2021

Edited by Max Alekseyev, Sep 10 2011

A068383 Numbers k such that k divides 11^k - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 16, 18, 20, 24, 25, 30, 32, 36, 40, 42, 48, 50, 54, 60, 64, 72, 80, 84, 90, 96, 100, 108, 114, 120, 125, 126, 128, 144, 150, 156, 160, 162, 168, 180, 192, 200, 210, 216, 222, 228, 240, 244, 250, 252, 256, 270, 272, 288, 294, 300, 312, 320

Offset: 1

Benoit Cloitre, Mar 05 2002

For all k, 2^k, 10^k, 2 * 3^k and 10 * 3^k are in the sequence.

			11^5 - 1 = 161050, which is divisible by 5, so 5 is in the sequence.
11^6 - 1 = 1771560, which is divisible by 6, so 6 is in the sequence.
11^7 = 19487171 = 4 modulo 7, so 7 is not in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014945, A014946, A014949, A014950, A068382.

Join[{1}, Select[Range[500], PowerMod[11, #, #] == 1 &]] (* Robert Price, Apr 04 2020 *)

isok(n) = Mod(11, n)^n == Mod(1, n); \\ Michel Marcus, May 06 2016

def powerMod(a: Int, b: Int, m: Int): Int = b match { case 1 => a % m; case n => a * powerMod(a, n - 1, m) % m }
List(1) ++: (2 to 500).filter(k => powerMod(11, k, k) == 1) // Alonso del Arte, Apr 04 2020

A333432 A(n,k) is the n-th number m that divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 9, 8, 0, 6, 1, 5, 4, 21, 16, 0, 7, 1, 2, 25, 6, 27, 20, 0, 8, 1, 7, 3, 125, 8, 63, 32, 0, 9, 1, 2, 49, 4, 625, 12, 81, 40, 0, 10, 1, 3, 4, 343, 6, 1555, 16, 147, 64, 0, 11, 1, 2, 9, 8, 889, 8, 3125, 18, 171, 80, 0, 12

Offset: 1

Seiichi Manyama, Mar 21 2020

			Square array A(n,k) begins:
  1, 1,  1,   1,  1,     1,  1,     1,  1, ...
  2, 0,  2,   3,  2,     5,  2,     7,  2, ...
  3, 0,  4,   9,  4,    25,  3,    49,  4, ...
  4, 0,  8,  21,  6,   125,  4,   343,  8, ...
  5, 0, 16,  27,  8,   625,  6,   889, 10, ...
  6, 0, 20,  63, 12,  1555,  8,  2359, 16, ...
  7, 0, 32,  81, 16,  3125,  9,  2401, 20, ...
  8, 0, 40, 147, 18,  7775, 12,  6223, 32, ...
  9, 0, 64, 171, 24, 15625, 16, 16513, 40, ...

Seiichi Manyama, Antidiagonals n = 1..25, flattened
OEIS Wiki, 2^n mod n

Columns k=1-20 give: A000027, A063524, A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A333506, A128360.
Main diagonal gives A333433.
Cf. A333429, A333500.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=2 then `if`(n=1, 1, 0) else
        while nops(p(k)) 1 do od;
          p(k):= [p(k)[], h]
        od; p(k)[n] fi
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Mar 24 2020

A[n_, k_] := Module[{h, p}, p[_] = {1}; If[k == 2, If[n == 1, 1, 0], While[ Length[p[k]] < n, For[h = 1 + p[k][[-1]], Mod[k^h, h] != 1, h++]; p[k] = Append[p[k], h]]; p[k][[n]]]];
Table[A[n, 1+d-n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A232769 Numbers n not divisible by 9 such that n divides 10^n - 1 (A014950).

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A127100 Numbers k such that k^2 divides 10^k-1.

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A177243 Numbers k such that k^3 divides 10^(k^2) - 1.

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A014960 Integers n such that n divides 24^n - 1.

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A014956 Positive integers k such that k divides 14^k - 1.

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A066364 Prime divisors of solutions to 10^n == 1 (mod n).

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A068382 Numbers k such that k divides 9^k - 1.

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A014957 Positive integers k that divide 16^k - 1.

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